Without wishing to suggest that professional philosophers would regard the book as philosophy, I can report that this book is definitely philosophical. Most of the book pertains to mathematical invention, but not just the psychology thereof, with many examples of the way in which mathematical advances move from two different and incompatible ways of viewing something to a higher viewpoint on it that makes better sense and better mathematics. A simple example of this is the invention of zero, where the (...) two incompatible viewpoints are that numbers are for counting and that there is nothing to count. The number one exemplified almost the same degree of blockage for the ancient Greeks, for whom the least number was two. It is perhaps unfortunate that the word that the author chose to represent the presence of such resolvable cognitive difficulties is ‘ambiguity’. As ambiguity is severely shunned by mathematicians and as there is none of it—as the word is normally used—in such situations as are described either before, when there are the two viewpoints, or later when there is a higher one, the use of ‘ambiguity’ would be misleading if it were not so adequately explained not to mean ambiguity. The excuse for using the word is claimed to be the genuine ambiguity of one of the simplest examples discussed, 3 + 4, with indifferently the meanings ‘add four to three’ and …. (shrink)

The study that George Lakoff and Rafael Núñez call "idea analysis" and begin in their recent book Where mathematics comes from is intended to dissect mathematical concepts into their metaphorical parts, where metaphor is used in the cognitive-science sense promoted by Lakoff and Mark Johnson in Metaphors we live by and subsequent works by each of them and together. Lakoff and Núñez's analysis of the (modern) algebraic concept of group is based on the attribution to contemporary mathematics of what will (...) be widely recognizable by their name for it, the folk theory of essences. I argue that this philosophical basis for their analysis is spurious and supply an alternative analysis of the same concept within their "metaphorical" paradigm but without essences. This analysis, which I hope is more viable than theirs, is intended to support the general applicability of the paradigm by freeing it from outmoded philosophical baggage. (shrink)